It sounds like one of the door hinge cams is broken or cracked.
You did not say which door so I listed both.

Item 22A in Section 1 for the freezer door (I think; but am guessing as there is no picture of it to confirm that this is the cam).
Item 22 in Section 2 for the fresh food door.
Item 22 in Section 3 for the lower hinge, looks the same for both doors.

Yes remove the top hinge.
Empty out the door to start with.

If the door has wires etc. going into it I find the following the easiest way to do it though it takes two people.
Get 2X6 6 or 2X4 or whatever you have kicking around.
Built up a stand that is about the same height as the hinge.
You the other person can then hold the door on the stand while yo do the repair.
You do not want to be stressing any wires or water hook ups.